<title>Triplet Markov chains in hidden signal restoration</title>

Abstract

Hidden Markov Chain (HMC) models are widely used in various signal or image restoration problems. In such models, one considers that the hidden process X=(X1, ., Xn) we look for is a Markov chain, and the distribution p(y/x) of the observed process Y=(Y1, ., Yn), conditional on X, is given by p(y/x)=p(y1/x1). p(yn/xn). The 'a posteriori' distribution p(x/y) of X given Y=y is then a Markov chain distribution, which makes possible the use of different Bayesian restoration methods. Furthermore, all parameters can be estimated by the general 'Expectation-Maximization' algorithm, which renders Bayesian restoration unsupervised. This paper is devoted to an extension of the HMC model to a 'Triplet Markov Chain' (TMC) model, in which a third auxiliary process U is introduced and the triplet (X, U, Y) is considered as a Markov chain. Then a more general model is obtained, in which X can still be restored from Y=y. Moreover, the model parameters can be estimated with Expectation-Maximization (EM) or Iterative Conditional Estimation (ICE), making the TMC based restoration methods unsupervised. We present a short simulation study of image segmentation, where the bi- dimensional set of pixels is transformed into a mono-dimensional set via a Hilbert-Peano scan, that shows that using TMC can improve the results obtained with HMC.